Note: This definition implies that repeating decimals are irrational number.
Let's list a common magic proof in the way as a brief explanation:
(1) x= 0.999...
(2) 10x= 9+x // 10x= 9.999...
(3) 9x=9
(4) x=1
Ans: There is no axiom or theorem to prove (1) => (2).
The purpose this text is for establishing the bases for computational algorithm.
This file https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
| Real Number |
+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a plus/minus sign or a point:
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ]
<dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
Two n-ary fixed-point number x,y are equal iff their form as mentioned above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
On 24/03/24 15:59, wij wrote:
The purpose this text is for establishing the bases for computational algorithm.
This file https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the >>> string may contain a plus/minus sign or a point:
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ]
<dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
Two n-ary fixed-point number x,y are equal iff their form as mentioned above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
So the RATIO of 1 and 7 isn't RATIOnal?
All rationals p/q are representable by q-ary fixed-point number.
On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
On 25/03/24 00:05, wij wrote:
On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
On 24/03/24 15:59, wij wrote:
The purpose this text is for establishing the bases for computational algorithm.
This file https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the >>>>> string may contain a plus/minus sign or a point:
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ]
<dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
Two n-ary fixed-point number x,y are equal iff their form as mentioned above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
So the RATIO of 1 and 7 isn't RATIOnal?
All rationals p/q are representable by q-ary fixed-point number.
0.99999... is representable by the following 10-ary fixed-point number: 1.0.
How?
On Mon, 2024-03-25 at 01:59 +0100, immibis wrote:
On 25/03/24 01:53, wij wrote:
On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
On 25/03/24 00:05, wij wrote:
On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
On 24/03/24 15:59, wij wrote:
The purpose this text is for establishing the bases for computational algorithm.
This file
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a plus/minus sign or a point:
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ] >>>>>>> <dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
Two n-ary fixed-point number x,y are equal iff their form as mentioned above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
So the RATIO of 1 and 7 isn't RATIOnal?
All rationals p/q are representable by q-ary fixed-point number.
0.99999... is representable by the following 10-ary fixed-point number: 1.0.
How?
By the proof you already showed and rejected, they are two ways of
writing the same number. At least in the real numbers. If you make up a
new number system, it won't be the same as the real numbers and you
shouldn't call it the real numbers.
It not 'make-up', instead, your math. theory is fabricated, in way worst than mine, at least.
If you like to stick to the theory that the universe revolves around the earth, that is fine.
One thing to be sure, such theory won't survive.
On Mon, 2024-03-25 at 03:29 +0100, immibis wrote:
On 25/03/24 02:31, wij wrote:
On Mon, 2024-03-25 at 01:59 +0100, immibis wrote:
On 25/03/24 01:53, wij wrote:
On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
On 25/03/24 00:05, wij wrote:
On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
On 24/03/24 15:59, wij wrote:
The purpose this text is for establishing the bases for computational algorithm.
This file
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a plus/minus sign or a point: >>>>>>>>>
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ] >>>>>>>>> <dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
Two n-ary fixed-point number x,y are equal iff their form as mentioned above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
So the RATIO of 1 and 7 isn't RATIOnal?
All rationals p/q are representable by q-ary fixed-point number. >>>>>>>
0.99999... is representable by the following 10-ary fixed-point number: 1.0.
How?
By the proof you already showed and rejected, they are two ways of
writing the same number. At least in the real numbers. If you make up a >>>> new number system, it won't be the same as the real numbers and you
shouldn't call it the real numbers.
It not 'make-up', instead, your math. theory is fabricated, in way worst than mine, at least.
If you like to stick to the theory that the universe revolves around the earth, that is fine.
One thing to be sure, such theory won't survive.
If two real numbers defined by Dedekind cuts are different, there is a
number in between them. This is obvious from the Dedekind cut
definition. If you invent a different system, you can do that but it's a
different system.
Let's say yes (if you'd like me to put this way), I am inventing a system to replace your obsolete
system.
On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
On 25/03/24 00:05, wij wrote:
On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
On 24/03/24 15:59, wij wrote:
The purpose this text is for establishing the bases for computational algorithm.
This file https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the >>>>> string may contain a plus/minus sign or a point:
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ]
<dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
Two n-ary fixed-point number x,y are equal iff their form as mentioned above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
So the RATIO of 1 and 7 isn't RATIOnal?
All rationals p/q are representable by q-ary fixed-point number.
0.99999... is representable by the following 10-ary fixed-point number: 1.0.
How?
On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:
Op 25.mrt.2024 om 01:53 schreef wij:What is that? Refutation by assertion?
On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
On 25/03/24 00:05, wij wrote:
On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
On 24/03/24 15:59, wij wrote:
The purpose this text is for establishing the bases for computational algorithm.
This file
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a plus/minus sign or a point:
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ] >>>>>>> <dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
Two n-ary fixed-point number x,y are equal iff their form as mentioned above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
So the RATIO of 1 and 7 isn't RATIOnal?
All rationals p/q are representable by q-ary fixed-point number.
0.99999... is representable by the following 10-ary fixed-point number: 1.0.
How?
1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
If not (if there is a difference), which number expresses the difference
between 1.0 and 0.99999... ?
Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and b. (dense property)
Op 25.mrt.2024 om 04:11 schreef wij:
On Mon, 2024-03-25 at 03:29 +0100, immibis wrote:
On 25/03/24 02:31, wij wrote:
On Mon, 2024-03-25 at 01:59 +0100, immibis wrote:
On 25/03/24 01:53, wij wrote:
On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
On 25/03/24 00:05, wij wrote:
On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
On 24/03/24 15:59, wij wrote:
The purpose this text is for establishing the bases for computational
algorithm.
This file
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a plus/minus sign or a point: >>>>>>>>>>
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ] >>>>>>>>>> <dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys
depending on n-ary
Two n-ary fixed-point number x,y are equal iff their form as
mentioned above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are
irrational number.
So the RATIO of 1 and 7 isn't RATIOnal?
All rationals p/q are representable by q-ary fixed-point number. >>>>>>>>
0.99999... is representable by the following 10-ary fixed-point number: 1.0.
How?
By the proof you already showed and rejected, they are two ways of
writing the same number. At least in the real numbers. If you make up a >>>>> new number system, it won't be the same as the real numbers and you
shouldn't call it the real numbers.
It not 'make-up', instead, your math. theory is fabricated, in way
worst than mine, at least.
If you like to stick to the theory that the universe revolves around
the earth, that is fine.
One thing to be sure, such theory won't survive.
If two real numbers defined by Dedekind cuts are different, there is a
number in between them. This is obvious from the Dedekind cut
definition. If you invent a different system, you can do that but it's a >>> different system.
Let's say yes (if you'd like me to put this way), I am inventing a
system to replace your obsolete
system.
A system becomes obsolete only if a new system has been worked out and
it is proved to have an advantage.
1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
If not (if there is a difference), which number expresses the
difference between 1.0 and 0.99999... ?
+-------------+
| Real Number |
+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a plus/minus sign or a point:
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ]
<dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending
on n-ary
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point
number. The string of digits of x may be infinitely long }
On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:
Op 25.mrt.2024 om 01:53 schreef wij:What is that? Refutation by assertion?
On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
On 25/03/24 00:05, wij wrote:
On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
On 24/03/24 15:59, wij wrote:
The purpose this text is for establishing the bases for computational algorithm.
This file
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a plus/minus sign or a point:
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ] >>>>>>> <dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
Two n-ary fixed-point number x,y are equal iff their form as mentioned above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
So the RATIO of 1 and 7 isn't RATIOnal?
All rationals p/q are representable by q-ary fixed-point number.
0.99999... is representable by the following 10-ary fixed-point number: 1.0.
How?
1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
If not (if there is a difference), which number expresses the difference
between 1.0 and 0.99999... ?
Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and b. (dense property)
IMO, these all boil down to the question "what is number?".
I find it better
(so far) to define 'number' on symbols and the associated ops.
Because I feel computation (TM, algorithm,program,..) can and should be a large
part of the basics of mathematics.
E.g. I feel infinity may be nothing but a
manifest of procedural loop.
As usual, you don't really care about the details, do you? It's all“Premature optimization is the root of all evil”
about looking technical rather than being precise.
On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:...
...1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
If not (if there is a difference), which number expresses the difference
between 1.0 and 0.99999... ?
Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and
b. (dense property)
On Mon, 2024-03-25 at 10:24 +0000, Ben Bacarisse wrote:
wij <wyniijj5@gmail.com> writes:Good catch. I will fix these grammar things, latter.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the >> > string may contain a plus/minus sign or a point:
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ]
<dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending
on n-ary
So neither 0 nor 10 are Fixed_point numbers. OK. That's going to limit >> the usefulness of these numbers.
dittoReal Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point
number. The string of digits of x may be infinitely long }
That's ill-defined as there are two {0, <nzd>} parts in the grammar. I
think you intended only one of them to be permitted to be not finite.
(And as has been pointed out, this is not a model of set to the realIMO, these all boil down to the question "what is number?".
numbers. You need to choose a different name.)
I find it
better (so far) to define 'number' on symbols and the associated ops.
Because I feel computation (TM, algorithm,program,..) can and should
be a large part of the basics of mathematics. E.g. I feel infinity may
be nothing but a manifest of procedural loop. (This article tries to
grasp this kind of idea https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download )
As usual, you don't really care about the details, do you? It's all“Premature optimization is the root of all evil”
about looking technical rather than being precise.
On Mon, 2024-03-25 at 07:14 -0400, Richard Damon wrote:
On 3/25/24 5:11 AM, wij wrote:
On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:
Op 25.mrt.2024 om 01:53 schreef wij:What is that? Refutation by assertion?
On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
On 25/03/24 00:05, wij wrote:
On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
On 24/03/24 15:59, wij wrote:
The purpose this text is for establishing the bases for computational algorithm.
This file
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a plus/minus sign or a point: >>>>>>>>>
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ] >>>>>>>>> <dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
Two n-ary fixed-point number x,y are equal iff their form as mentioned above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
So the RATIO of 1 and 7 isn't RATIOnal?
All rationals p/q are representable by q-ary fixed-point number. >>>>>>>
0.99999... is representable by the following 10-ary fixed-point number: 1.0.
How?
1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
If not (if there is a difference), which number expresses the difference >>>> between 1.0 and 0.99999... ?
Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and b. (dense property)
But the density property requires a and b to be DIFFERENT numbers before
it asserts that there is a number between them, otherwise c is just the
same number as a and b.
The premise of this thread said "If not" (means IF 0.999...!=1)
On Mon, 2024-03-25 at 19:43 -0400, Richard Damon wrote:
On 3/25/24 7:54 AM, wij wrote:
On Mon, 2024-03-25 at 07:14 -0400, Richard Damon wrote:
On 3/25/24 5:11 AM, wij wrote:
On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:
Op 25.mrt.2024 om 01:53 schreef wij:What is that? Refutation by assertion?
On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
On 25/03/24 00:05, wij wrote:
On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
On 24/03/24 15:59, wij wrote:
The purpose this text is for establishing the bases for computational algorithm.
This file
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
+-------------+
Real Number |+-------------+
n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a plus/minus sign or a point: >>>>>>>>>>>
<fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ]
<dstr1>::= <nzd> [{ 0, <nzd> } <nzd>] >>>>>>>>>>> <dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-
ary
Two n-ary fixed-point number x,y are equal iff their form as mentioned
above
are identical.
Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }
Note: This definition implies that repeating decimals are irrational
number.
So the RATIO of 1 and 7 isn't RATIOnal?
All rationals p/q are representable by q-ary fixed-point number. >>>>>>>>>
0.99999... is representable by the following 10-ary fixed-point number: 1.0.
How?
1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
If not (if there is a difference), which number expresses the difference >>>>>> between 1.0 and 0.99999... ?
Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and b. (dense property)
But the density property requires a and b to be DIFFERENT numbers before >>>> it asserts that there is a number between them, otherwise c is just the >>>> same number as a and b.
The premise of this thread said "If not" (means IF 0.999...!=1)
False premise -> Unsound conclusions.
If you want 0.999(9) to be different than 1.00 you need to move beyond
the Real Numbers into some form of extended Reals, and understand what
that does to what logic you can use.
If going into some alternate or extended number system IS your goal,
then you shouldn't call your numbers "The Reals", as that name is taken.
'real number' is already there. e.g. 'a line' or just a series of digits, whatever had practically used world wide now and then.
Your 'real number' is also a theory which also suffers from revision.
If you need a specific number system, you should name it otherwise.
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